Rút gọn biểu thức B =(x-1/x+1):(x+3/x-2+x+2/3-x+x+2/x^2+5x+6)

Đáp án:

\(\dfrac{{x – 1}}{{\left( {x + 1} \right)\left( {x – 2} \right)}}\)

Giải thích các bước giải:

\(\begin{array}{l}
B = \dfrac{{x – 1}}{{x + 1}}:\left[ {\dfrac{{x + 3}}{{x – 2}} + \dfrac{{x + 2}}{{3 – x}} + \dfrac{{x + 2}}{{{x^2} + 5x + 6}}} \right]\\
 = \dfrac{{x – 1}}{{x + 1}}.\left[ {\dfrac{{\left( {x + 3} \right)\left( {x – 3} \right) – \left( {x + 2} \right)\left( {x – 2} \right) + x + 2}}{{\left( {x – 2} \right)\left( {x – 3} \right)}}} \right]\\
 = \dfrac{{x – 1}}{{x + 1}}.\dfrac{{{x^2} – 9 – {x^2} + 4 + x + 2}}{{\left( {x – 2} \right)\left( {x – 3} \right)}}\\
 = \dfrac{{x – 1}}{{x + 1}}.\dfrac{{x – 3}}{{\left( {x – 2} \right)\left( {x – 3} \right)}}\\
 = \dfrac{{x – 1}}{{\left( {x + 1} \right)\left( {x – 2} \right)}}
\end{array}\)